| MATHEMATICS | |||||
| Bachelor | TR-NQF-HE: Level 6 | QF-EHEA: First Cycle | EQF-LLL: Level 6 | ||
| Course Code | Course Name | Semester | Theoretical | Practical | Credit | ECTS |
| MAT2004 | Linear Algebra and Analytic Geometry | Spring | 3 | 2 | 4 | 5 |
| Language of instruction: | English |
| Type of course: | Must Course |
| Course Level: | Bachelor’s Degree (First Cycle) |
| Mode of Delivery: | Face to face |
| Course Coordinator : | Instructor NERMINE AHMED EL SISSI |
| Course Lecturer(s): |
RA ESRA ADIYEKE |
| Recommended Optional Program Components: | None |
| Course Objectives: | This course is the continuation of the first linear algebra course, MAT2003. The course will emphasize abstract vector spaces and linear maps. In this course, students will develop their ability to understand and manipulate the objects of linear algebra. In addition, students will get exposed to the versatility of linear algebra in different branches of mathematics. |
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The students who have succeeded in this course; • Define and analyze finite and infinite dimensional vector space and subspaces over a field. • Build new vector spaces from old ones using direct sums. • Define and use the properties of linear transformations. • Find the null space, range and matrix representation of a given linear transformation. • Determine whether a linear transformation is one-to-one and onto. • Determine the diagonalizability of a linear transformation using eigenvectors and eigenvalues. • Define inner product and use the notion of inner products to define orthogonal vectors. • Apply the Gram-Schmidt process to generate an orthonormal set of vectors. • Identify normal, self-adjoint and positive operators. • Find the singular value decomposition of a matrix. • Compute the characteristic and minimal polynomials of an operator. • Compute the polar decomposition of a matrix. • Find the Jordan canonical form of matrices. • Define and apply properties of the trace and determinant of an operator and of a matrix. |
| The course covers the following topics: complex vector spaces, direct sums, linear maps products and quotients of vector spaces, dual spaces, invariant subspaces, quotient spaces, inner product spaces, orthonormal bases and Gram-Schmidt orthogonalization, self-adjoint and normal operators, the spectral theorem, positive operators and isometries, polar decomposition, singular value decomposition, characteristic and minimal polynomials of an operator, canonical forms, and the trace and determinant of an operator. |
| Week | Subject | Related Preparation |
| 1) | Vector Spaces, subspace and direct sum | |
| 2) | The vector space of linear maps, null spaces and ranges, matrix representation of a linear map | |
| 3) | Invertible linear map and isomorphic vector spaces, product of vector spaces | |
| 4) | Quotients of vector space, duality, | |
| 5) | Invariant subspaces, eigenvectors and upper-triangular matrices | |
| 6) | Eigenspace and diagonal matrices, and Inner product spaces and norms | |
| 7) | Orthonormal bases and Orthogonal complements | |
| 8) | self-adjoint and normal operators and The spectral theorem | |
| 9) | Positive operators and isometries, Polar Decomposition | |
| 10) | Singular Value Decomposition and operators on Complex vector spaces | |
| 11) | Decomposition of an operator, and The Cayley-Hamilton Theorem | |
| 12) | The minimal polynomial, and the Jordan form | |
| 13) | The determinant of an operator, the determinant of a matrix, volume | |
| 14) | Review |
| Course Notes / Textbooks: | Linear Algebra Done Right, Sheldon Axler, Springer, Third Edition. |
| References: | . |
| Semester Requirements | Number of Activities | Level of Contribution |
| Quizzes | 7 | % 10 |
| Presentation | 3 | % 5 |
| Midterms | 2 | % 40 |
| Final | 1 | % 40 |
| Paper Submission | 3 | % 5 |
| Total | % 100 | |
| PERCENTAGE OF SEMESTER WORK | % 60 | |
| PERCENTAGE OF FINAL WORK | % 40 | |
| Total | % 100 | |
| Activities | Number of Activities | Duration (Hours) | Workload |
| Course Hours | 14 | 3 | 42 |
| Application | 14 | 2 | 28 |
| Study Hours Out of Class | 14 | 3 | 42 |
| Presentations / Seminar | 3 | 3 | 9 |
| Quizzes | 7 | 1 | 7 |
| Midterms | 2 | 1 | 2 |
| Paper Submission | 3 | 3 | 9 |
| Final | 1 | 2 | 2 |
| Total Workload | 141 | ||
| No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
| Program Outcomes | Level of Contribution | |
| 1) | To have a grasp of basic mathematics, applied mathematics and theories and applications in Mathematics | 5 |
| 2) | To be able to understand and assess mathematical proofs and construct appropriate proofs of their own and also define and analyze problems and to find solutions based on scientific methods, | 5 |
| 3) | To be able to apply mathematics in real life with interdisciplinary approach and to discover their potentials, | 3 |
| 4) | To be able to acquire necessary information and to make modeling in any field that mathematics is used and to improve herself/himself, | 4 |
| 5) | To be able to tell theoretical and technical information easily to both experts in detail and non-experts in basic and comprehensible way, | 3 |
| 6) | To be familiar with computer programs used in the fields of mathematics and to be able to use at least one of them effectively at the European Computer Driving Licence Advanced Level, | 4 |
| 7) | To be able to behave in accordance with social, scientific and ethical values in each step of the projects involved and to be able to introduce and apply projects in terms of civic engagement, | 1 |
| 8) | To be able to evaluate all processes effectively and to have enough awareness about quality management by being conscious and having intellectual background in the universal sense, | 3 |
| 9) | By having a way of abstract thinking, to be able to connect concrete events and to transfer solutions, to be able to design experiments, collect data, and analyze results by scientific methods and to interfere, | 3 |
| 10) | To be able to continue lifelong learning by renewing the knowledge, the abilities and the competencies which have been developed during the program, and being conscious about lifelong learning, | 3 |
| 11) | To be able to adapt and transfer the knowledge gained in the areas of mathematics ; such as algebra, analysis, number theory, mathematical logic, geometry and topology to the level of secondary school, | 3 |
| 12) | To be able to conduct a research either as an individual or as a team member, and to be effective in each related step of the project, to take role in the decision process, to plan and manage the project by using time effectively. | 2 |